ON CHARACTERIZATIONS OF LINEAR GROUPS, II(X) BY MICHIO SUZUKI The main object of this paper is to characterize the linear fractional group Go in 2 variables over a finite field F of characteristic 2 by the properties of involutions in it. Here by an involution we shall mean an element of order 2. Go is isomorphic with the factor group of V, the totality of 3 X3 nonsingular matrices over F, modulo its center A. It is easily seen that any involution of T is conjugate to the involution 7: 10 0 7=010. .10 1. Let fl be the totality of 3X3 nonsingular matrices commuting with 7. Then _ is a group of order q3(q — l)2 consisting of triangular matrices with the bottom main diagonal element equal to the top one. Hence in Go the central- izer 7V"oof an involution has order qz(q —1) and is isomorphic with the factor group fl/A of Q modulo the center of V. Our main theorem is the converse to this statement. Theorem. Let G be a finite group of even order and r be an involution of G. If the centralizer of t is isomorphic with fl/A and if every involution of G is con- jugate to t, then G is isomorphic with Go, the linear fractional group of 2 vari- ables, with one exception. The exceptional case occurs when F has only two ele- ments and in this case we have G~LF(3, 2) or G=A6. A similar characterization of the linear fractional group in 2 variables over a field of odd characteristic has been obtained by R. Brauer in the case that the ground field has q elements with q= —1 (mod 4) (cf. [2; 2*]). 1. In this section we shall consider the group fi/A and derive several properties of this group which will be used in the subsequent argument. Throughout this section 7V0stands for the group _/A. It is clear that 7V0 is isomorphic with the totality of 3X3 matrices of the form 1 0 0' M(a, 0,y;8)= a 8 0 .13 7 1. Received by the editors February 10, 1958. 0) The investigation has been done at Harvard University with support from the National Science Foundation: NSF Grant, G 2268. 205 206 MICHIO SUZUKI [August where a, 8, y and 8^0 are elements of a finite field F of characteristic 2. We take No as the totality of M(a, B,y; 8). Let q be the number of elements con- tained in F so that q = 2". The matrix multiplication may be written as M(a, 8, y; h)M(a', 8', y'; 5') = M(a", 8", y"; 8") where a" = a + ba', 8" = 8 + ya' + 8', y" = yd' + y' and 8" = 88'. Hence the mapping cp of N0 defined by cp(M(a,B,y; 8)) = 8 is a homomorphism of No onto the multiplicative group of nonzero elements of F. The kernel of cp is the totality of M(a, B,y; 1) and is of course a 2-group of order q3. Thus we have the following proposition. (I) No contains a normal subgroup Q of order q3 and the factor group N0/Q is a cyclic group of order q — 1. Furthermore from the matrix multiplication we conclude the following propositions, which may be proved by computation. (II) M(a, 8, y; 1)2 = M(0, ay, 0; 1). In particular M(a, 8, y; 1) is an in- volution if ay = 0; otherwise the order is 4. (III) The center C of Q is of order q consisting of elements M(0, B, 0; 1), and C is the center of A70. (IV) The centralizer of each element n^l oi No with an order a divisor of q—1 is an abelian group of order q(q— 1). (V) The centralizer of each element of order 4 in A^0 is an abelian group of order q2. Let P be the subgroup of Q consisting of matrices M(a, B, 0; 1) and L be the subgroup corresponding to the totality of matrices M(0, 8, y; 1). (VI) Both P and L are abelian groups of order q2 consisting of elements of order =£2. They are normal in Q. (VII) Every involution of A^0is contained in either P or L and the inter- section of P and L is the center C. These two propositions are obvious from the property (II) and the defini- tion of P and L. (VIII) Involutions of P not contained in C are conjugate to each other in iVo. The same property holds for L. Proof. We have M(0, 0, X; 8~l)-1 = M(0, 0, 8X; 8) and M(0, 0, X; 8~1)M(a, 8, 0; 1)M(0, 0, SA;8) = M(8~xa, \a + 8, 0; 1). Hence if M(a, 8, 0; 1) is not in C then a\^0 and we may take X and 8 in such a way that 8 = a and \a+B = 0. Hence every involution of P not contained in C is conjugate to M(l, 0, 0; 1) and this is the statement (VIII). Since Q = pyJL, Q/C is a direct product of P/C and L/C. (IX) The factor group Q/C is an abelian group of order q2 and of type (2, • • • , 2). 1959] ON CHARACTERIZATIONS OF LINEAR GROUPS, II 207 Finally we shall prove the following proposition. (X) The commutator subgroup of 7V0is Q if q>2. Since N0/Q is a cyclic group the commutator subgroup 7V0' of No is a part of Q. Since M(a-1, 8, 0; 1)M(1, 0, 0; l) = M(a, B, 0; 1), we see that if a^l and 0, M(a, 8, 0; 1) is a commutator in 7V0 (cf. the computation in (VIII)). If g>2, the field F contains an element a which is neither 0 nor 1. Then M(a, 8,0; 1) is an involution of P which is not in Cand is a commutator. Hence by (VIII) every involution of P not contained in C is a commutator. Thus the commutator subgroup 7V~0'contains P. Similarly L is a part of 7V0' and hence Q = PULQN{. 2. Let G be a finite group of even order, and t be an involution of G. In the rest of this paper we assume the two conditions (A) and (B): (A) the centralizer No of r in G is isomorphic with fi/A; (B) every involution of G is conjugate to t in G. We shall use the same notations introduced in the first section; i.e. C stands for the center of 7V0, Q the 2-Sylow subgroup of 7V0and P or L is the subgroup of Q as defined in the §1. Proposition 1. Q is a 2-Sylow subgroup of G. Proof. By way of contradiction suppose that Q is a proper subgroup of a 2-Sylow subgroup Q* oi G. We can take a subgroup Qj of Q* such that [Q': (?] = 2. By a property of 2-groups Q is a normal subgroup of Q'. Since C is the center of Q, it is a characteristic subgroup of Q. Hence C is a normal subgroup of Q'. Again by a property of 2-groups we see that C contains a central element t't^I of Q'. By the assumption (B) t' is conjugate to r in G and hence the centralizer of t' is isomorphic with 7V0. Since t' is in the center of Q' we conclude that a 2-Sylow subgroup of the normalizer of t' has a greater order than Q. This is impossible. Proposition 2. P is never conjugate to L in G. Proof. By way of contradiction suppose that there is an element irQG such that 7rPir_1 = L. Then irQir-1 and Q are 2-Sylow subgroups of the normal- izer of L by (VI) and Proposition 1. Hence there is an element p such that pLp~x = L and pirQir~1p~l = Q. Put a = pir. We have aPa~x = pirPir^p"1 = pLp-1 = L and aQcr~1 = Q. Hence aLa*1 is a subgroup of Q consisting of elements of order ^2. By the property (VII) of the first section aLa~1 = P. Hence some power ai of <r has an order a power of 2 and still exchanges P and L. Q and ai generate a 2-group. By Proposition 1 Q is a 2-Sylow subgroup of G and hence Q contains ai. Since both P and L are normal subgroups of Q by (VI) of §1, we get a contradiction. 3. If q = 2, the group 7V0is a dihedral group of order 8. Hence 7V0contains an element ir of order 4 such that t=7t2. Since 7V0 is the centralizer of t in G, the centralizer of ir in G is a part of 7V"oand hence coincides with the cyclic 208 MICHIO SUZUKI [August group generated by ir. The structure of a finite group possessing such a property is known (cf. [7]). From the results of [7] we conclude that Gis isomorphic with either LF(2, 7)^LF(3, 2) or LF(2, 9)^At, the alternating group on six letters. In fact a direct proof of this result can be carried through by using a method similar to the one in [7] and it is much easier.